DTN Meteo: A Generic Framework for Predicting DTN Protocol PerformanceDTN Protocol Performance

The previous approaches are able to provide good approximations or upper bounds for the performance of epidemic routing, and simple variants. The common denominator of all the protocol variants is that they are “random”, in the sense that none of them ex-plicitly differentiates among nodes, based on their characteristics (mobility, resources or other), when making a forwarding/dropping decision. In contrast, almost all newer DTN algorithms and protocols [35, 169, 170] base their decisions on nodes’ utilities with respect to the scheme’s end goal (position in the network, relationship to desti-nation etc). These node utilities are most often based on the observation that human mobility has non-trivial structure, manifesting in mixing/meeting patterns which can be extracted from real world experiments [171, 172].

We propose such a common analytical framework, which we entitle DTN-Meteo, and which is based on the followingthree common featuresof most DTN protocols.

Firstly, the bulk of proposed algorithms, whether for routing, content dissemination, distributed caching etc, essentially solve a combinatorial optimization problem over the state of every node in the network. Each algorithm defines a preference (utility) function over possible network states and aims at moving to better states. Secondly, candidate new states, in the DTN context, are presented according to thestochastic mobility of the nodes involved. As such,the traversal of the solution space of a problem is also stochastic. And thirdly, due to the difficulty (usually impossibility), in this context, of updating nodes’ states globally, protocols resort tolocal searchheuristics (with a deterministic or randomized search strategy) to choose between the current and a new possible state,involving state changes only in the two nodes in contact.

Using the above insight, DTN-Meteo maps a combinatorial optimization problem into a Markov chain (MC), where each state is a potential solution (e.g. assignment of content replicas to nodes), and transition probabilities are driven by two key factors: (i) node mobility, which makes available new potential solutions to the algorithm, over time, based on the (heterogeneous) contact probability of each node pair, and (ii)the algorithm, which orders the states using a utility function and accepts better ones either deterministically (greedy algorithm) or probabilistically (randomized algorithm), to avoid getting stuck in local maxima⁶. This not only decouples the algorithm’s effect and the mobility’s effect from each other, but also enables deriving useful performance metrics (convergence delay, delivery probability) using transient analysis of this MC.

LetNbe our Opportunistic Network, with|N |=Nnodes.N is a relatively sparse ad hoc network, where node density is insufficient for establishing and maintaining (end-to-end) multi-hop paths, in the sense of [174]. Instead, data is stored and carried by nodes, and forwarded through intermittent contacts established by node mobility. A contactoccurs between two nodes who are in range to setup a bi-directional wireless link to each other.

We assume an optimization problem over theN-node networkN (e.g. multicast under resource constraints) and a distributed algorithm for this problem, run by all

6In the spirit of solutions like Markov Chain Monte Carlo optimization and simulated annealing, for traditional optimization problems [173].

nodes. Our long term aim is to understand the performance of the algorithm as a func-tion of the nodes’ behavior and attributes (mobility, collaborafunc-tion, resource availability etc). Table 4.5 summarizes this section’s notation.

N andN the network (set of nodes), resp. its size i,j,k nodes inN

L maximum allowed replication (number of copies) SandΩ node, resp. network state space

x,y,z network states (elements ofΩ)

U(x),u(i) utility of a network state, resp. of a node δ(x,y) difference between network statesxandy Axy acceptance probability for transitionxy p^c_ij contact probability between nodesiandj P={pxy} Markov transition probability matrix

Tm^(ij),Tn starting times for(i, j)contacts , resp. any contacts Jm^(ij),Jn intercontact times for(i, j), resp. any contacts K^(ij)(t) residual intercontact times for(i, j)

αij power law exponent of(i, j)intercontact times βij rate parameter of(i, j)intercontact times Fij,Fij CDF, resp. CCDF of(i, j)intercontact times

Table 4.5: Important notation for Section 4.8.

4.8.1 Solution Space

We consider a class of DTN problems for which a solution can be expressed in terms of nodes’ states. In this paper, we restrict ourselves tobinary node states, to better illustrate the key concepts; however, in a more realistic variant of DTN-Meteo, nodes’

states could be chosen from a set ofB-bit integers(e.g. to allow the modeling of some storage constraint ofC 6B messages for each node, out of a maximumB contem-poraneous messages in the network). In all cases, the space of candidate solutions for such problems is a set ofN-element vectors, possibly restricted by a number of con-straints. Finally, an algorithm for the problem defines a ranking over these solutions, captured by a utility functionU(·). The goal is to maximize this utility (or minimize a cost function). We define our class of problems as follows:

• node state space S={0,1}orS ⊂N (4.6)

• network state space Ω⊆ S^N

Ω ={x|x= (x1, x2, . . . , xN)}, xi∈ S, ∀i∈ N (4.7)

• a set ofconstraints fi(x1, . . . , xN)6ρi (4.8)

• autility function U: Ω7→R (4.9)

This is, in fact, the combinatorial optimization class, which naturally encompasses several DTN problems, as they are dealing with indivisible entities (nodes, messages, channels etc) and have rules that define a finite number of allowable choices (choice of relays, assignment of channels etc). Below are some examples of DTN problems that can be thus modeled.

Content placement. The goal in content placement, is to make popular content (news, software update etc) easily reachable by interested nodes. As flooding the con-tent is unscalable, a small numberLof replicas can be pushed from its source to L strategic relays, which will store it for as long as it is relevant, and from whom en-countered interested nodes retrieve it⁷. In its simplest form, the source of the content distributes the L replicas toL initial nodes (e.g. randomly, using binary or source spraying [175]). These initial relays then forward their copies only to nodes that im-prove the desired utility – which can be based on mobility properties, willingness to help, resources, etc. (see [170] for some examples).

Here, the binary state of a nodeiis interpreted as the node being (xi = 1) or not being (xi = 0) a provider for the content of interest⁸. There is a single constraint for any allowed solution, namely∑N

i=1xi6L, that is, in Eq. (4.8)ρi=ρ=L.

Routing.In routing, be it unicast, multicast, broadcast, or anycast, the binary state of nodeiis interpreted as carrying or not carrying a message copy. For example, for unicast routing from source nodesto destination noded, the initial network state is x0 = (0,0, . . . ,{xs = 1}, . . . ,{xd = 0}, . . . ,0). The desired network state is any x^⋆, with xd = 1andxi ∈ {0,1}, ∀i ̸= d. This can easily be extended to group communication, with multiple sources and/or multiple destinations.

Various replication strategies can be expressed using Eq. (4.8) constraints, as shown in Table 4.6. Different schemes in each category, essentially differ in the utility function used to rank states and the action taken given a utility difference. In this paper, we analyze the first two strategies in Table 4.6.

Replication Constraint

i) single-copy [15, 12, 35] ∑N

i=1x_i= 1 ii) fixed budgetL [175, 147] ∑N

i=1x_i6L

iii) epidemic [11] ∑N

i=1xi6N iv) utility-based⁹ [105, 176, 169] ∑N

i=1xi6N Table 4.6: Modeling replication with Eq. (4.8)

As a final note, replication-based schemes have an initial spreading phase, during which theLcopies are distributed to the initial relays. This phase can straightforwardly be included in our model, by considering also all states with1toL−1copies in the network. This, however, increases the complexity of the model, while offering little added value. Indeed, the delay of this initial phase is negligible in networks with large N andL ≪ N, which is the case in scenarios of interest (see e.g. [175]). Hence, we will only consider the effect of this phase on the starting configuration (as shown in Section 4.9.1, Eq. (4.25)) of the algorithm and ignore its delay. Thus, our solution

7Lmust be derived to achieve a trade-off between desired performance and incurred costs, for example as in [175].

8AB-bit integer node state extends this to providingBpieces of content.

9Replicating only to higher utility nodes limits copies implicitly, but in the worst case, there will still be Ncopies.

spaceΩ, will only be composed of network states with exactlyLcopies atLdifferent nodes.

4.8.2 Exploring the Solution Space

In traditional optimization problems, local search methods define a neighborhood around the current solution, evaluate all solutions in the neighborhood and “move” towards the best one therein (purely greedy algorithms). Occasionally, they may also move to lower utility states (using randomization) in order to overcome local maxima, as in random-ized local search algorithms (e.g. simulated annealing). This aspect is fundamentally different in DTN optimization. The next candidate solution(s) cannot usually be cho-sen. Instead, the solution spaceΩis explored via node mobility (contacts): a new potential solution will only be offered, after some time, when two nodes come in con-tact. This has two major implications: (i) this new solution can differ (from the current) in the state of at most two nodes: the ones involved in the contact; (ii) which pair of nodes meets next (a random event) will define which potentialnewstate will become available, and how much better/worse the new state will be (whether the transition to the new state occurs, is finally decided by the algorithm). As a result, the traversal of the solution space is inherentlystochastic.

Consider implication (i) first (we treat (ii) when defining our network model, in Section 4.8.4). Every contact between a relay node/content provideri(x_i = 1) and another nodej(x_j = 0) offers the chance of moving to a new network statey∈ Ω, withy_i= 0andy_j= 1(forwarding) ory_i= 1andy_j= 1(replication).Ifthe replica is transferred from relay itoj, then the xy transition happens. Fig. 4.12 provides examples ofpotentialstate transitions, along with contacts required for the transitions to be possible.

Transitionxzin Fig. 4.12 requires two contacts tostartsimultaneously, and is thus not possible¹⁰. This means that only transitions betweenadjacent states are possible, where we define state adjacency as:

δ(x,y) = ∑

16i6N

I{xi̸=yi}62, (4.10)

i.e., the two network states may only differ in one or two nodes (I{x_i̸=y_i}are binary indicator variables). An encounter of those two nodes is a necessary (but not sufficient) condition for transitionxyto happen.

Summarizing, our solution space exploration goes as follows: When at a statex, the next contact between two nodesi, j presents a new potential solution y to the algorithm with a (time-homogeneous) probabilityp^c_ij, the contact probability of node pair(i, j)(we look deeper into these contact probabilities and the assumptions behind them in Section 4.8.4). The new solutionydiffers fromxin positionsiandjonly. For example, in Fig. 4.12(a): when at statex= (0,0,0,1,1,1)the algorithm could move to a new solutiony= (0,1,0,0,1,1)in the next contact, with probabilityp^c₂₄.

10We emphasize that this is not a strict necessity for our networks of interest, but rather a technical as-sumption of our mobility model, for analytical convenience (see Section 4.8.4 for more details). However, given the relative sparsity of the networks in question, such events occur with low probability.

Contact: Contact:

Contact: Contact:

(a) Forwarding

Contact: Contact:

(b) Replication

Figure 4.12: Example state transitions in two6-node DTNs.

4.8.3 Modeling a Local Optimization Algorithm

Node contacts merelyproposenew candidate solutions. Whether the relayidoes in fact hand over its message or content replica toj (or, more generally, whether a new allocation of objects betweeniandjis chosen), is decided by thealgorithm. In purely greedy (utility-ascent) schemes, apossiblestate transition occurs only if it improves the utility functionU, specific to each problem. Then, for our DTN problems, a possible transitionxyoccurs with binaryacceptance probability:

Axy =I{U(x)< U(y)}. (4.11) More generally, the acceptance probability may be any function of the two utilities:

Axy ∈[0,1]. This allows DTN-Meteo to model randomized local search algorithms (e.g. Markov Chain Monte Carlo methods, simulated annealing) as well.

It is important to make some remarks about utilities here. First, the (global) utility values above may sometimes require knowning the state of remote nodes (in the worst case, all nodes). This state may not be readily available locally at each node. Therefore, the algorithm must solve the added problem of collecting information and/or estimating it – usually through an integrated sampling component [105, 176]. Most DTN protocols suppose the existence of a mobility-related node utility u : N 7→ R(e.g. contact frequency). Thus, if the mobility is stationary and the estimation designed correctly, the online and offline utility rankings will coincide. However, estimating node properties and utilities is beyond the scope of this work.

Second, in anydistributedalgorithm, a node must be able to evaluatelocallywhether a given decision (e.g. to hand over its copy to an encountered node, for routing) will improve the global utility. This is a challenge for the algorithm designer, often solved by usingadditiveglobal utilities (e.g.U(x) =∑N

i=1x_i·u(i), whereu:N 7→Rcan be applied to each node separately). Therefore, DTN-Meteo assumes any state utility U and/or node utilityuare readily available.

Finally, we note that some protocols use packet-based utilities [176], while in DTN-Meteo we assume node-based utilities. Furthermore, the types of packet utili-ties in [176] are time-dependent. While such time-dependent packet utiliutili-ties cannot be modelled by our framework (as explained immediately below), some types of static

packet-based utilities could be included in future work.

Summarizing from sections 4.8.1, 4.8.2, and 4.8.3, the transition probability be-tweenadjacent network states xandy can be expressed in function of the contact probability (to be discussed in Section 4.8.4) and the acceptance probability as fol-lows:

p_xy =p^c_ij·A_xy, (4.12)

where nodesiandjare the two nodes whose encountercouldprovoke the state tran-sition.p^c_ij is themobility componentof the transition probability andA_xyis the algo-rithmic component.

We must stress here that Eq. (4.12) assumes that both variables are time-invariant, such that the resulting Markov chain for the problem will be time-homogeneous. Specif-ically, this requirement on the algorithm componentA_xy implies time-invariant (or very slowly varying) utilities, such as ones derived from social relationships among humans carrying the devices (nodes) which form our network. Additional types of utilities that are time-invariant (or very slowly varying), that DTN-Meteo can support, include functions based on computational power, operating system, trust relationships tied to social closeness, long-term traffic patterns etc.

The mobility componentp^c_ij (contact probability betweeni, j) is also assumed to be time-homogeneous. This is not unreasonable, as the above probabilities refer to long-term contact relations of humans, which tend to be stable. Some diurnal patterns might be present, but these could be further handled as discussed in the Section 4.8.6.

Therefore, for any two statesxandy, our algorithm is a discrete-time time-homogeneous Markov chain(X_n)_n∈N

0 over the solution spaceΩ, described by the transition proba-bility matrixP={p_xy}, with:

This formulation allows us to transform any problem of our defined class into a sim-ple Markov chain, which can be used for performance analysis and prediction. Fig. 4.13 presents the necessary steps to apply DTN-Meteo.

4.8.4 Modeling Heterogeneous Node Contacts

Recall our Opportunistic NetworkN, with|N |=N nodes andE 6(_N

2

)node pairs contacting each other. As data is exchanged only upon contacts inN, a mobility model based on contact patterns is sufficient for our analysis.

Definition 4.8.1(Node pair processes). Every node pair’s contact process is an inde-pendent, stochastic,stationary, and ergodic renewal process

( Tm^(ij)

)

m∈Z, whereTm^(ij)

denotes the starting time of a contact between nodes (i, j). The random variables